Parametrized complexity of relations between multidimensional subshifts

TL;DR

We address the parametrized complexity of central relations between multidimensional subshifts by fixing a parameter $Y$ and varying $X$, across SFTs and effective subshifts in dimension $d\ge 1$. Our approach uses reductions to classic computability problems (e.g., $\Halt$, $Total$, $COF$, $DP$) and explicit SFT constructions to characterize how dynamical properties of $Y$ influence decidability and hardness for relations such as $X=Y$, $X\simeq Y$, $X\subseteq Y$, $X\hookrightarrow Y$, $Y\subseteq X$, and $Y\hookrightarrow X$. We establish decidability criteria (e.g., $Y\subseteq X$ is decidable iff $\mathcal{L}(Y)$ is computable) and tight complexity bounds (ranging from $\Sigma^0_1$ to $\Sigma^0_3$, $\Pi^0_2$, and $D(\Sigma^0_1)$), with maximal complexities reachable for suitable parameter choices; we also connect conjugacy complexity to minimality and computable language, and show one-dimensional results can inform higher-dimensional cases via lifts. The results reveal sharp asymmetries across relations and highlight how parameter choices dramatically reshape the landscape of decidability in multidimensional symbolic dynamics. The work provides a nuanced picture of Rice-type undecidability phenomena in this setting and points to future avenues for a complete characterization of decidable properties for SFTs and effective subshifts.

Abstract

We study the parametrized complexity of fundamental relations between multidimensional subshifts, such as equality, conjugacy, inclusion, and embedding, for subshifts of finite type (SFTs) and effective subshifts. We build on previous work of E. Jeandel and P. Vanier on the complexity of these relations as two-input problems, by fixing one subshift as parameter and taking the other subshift as input. We study the impact of various dynamical properties related to periodicity, minimality, finite type, etc. on the computational properties of the parameter subshift, which reveals interesting differences and asymmetries. Among other notable results, we find choices of parameter that reach the maximum difficulty for each problem; we find nontrivial decidable problems for multidimensional SFT, where most properties are undecidable; and we find connections with recent work relating having computable language and being minimal for some property, showing in particular that this property may not always be chosen conjugacy-invariant.

Figures (1)

• Figure 1: An SFT that simulates a Turing machine $M$ on a free input, where $M$ has alphabet $\Sigma$ with blank symbol $B$, set of states $Q$, transition function $\delta$ and initial state $q_0$. The tiles are the elements of the alphabet, and adjacent edges with non-matching labels are forbidden. It is straightforward to modify the tiles to enforce an empty input or any fixed finite input. This procedure of associating an SFT to a Turing machine originates in Hao Wang's proof that the origin constrained version of the domino problem is undecidable wang_proving_1961, and the reader is referred to jeandel_undecidability_2020 for a more detailed exposition of the same construction.

Theorems & Definitions (79)

• Definition 2.1
• Definition 2.2
• Definition 2.3
• Definition 2.4
• Definition 2.5
• Proposition 2.6: Theorems 2.4.2, 4.3.2 and 4.3.3 in soare_turing_2016
• Proposition 2.7
• Definition 3.1
• Theorem 3.2
• Corollary 3.3